Finding Domain Of A Log Function

Finding the domain of a logarithmic function might seem daunting at first, but it becomes straightforward once you understand the underlying principles. Logarithmic functions are closely tied to exponential functions, and understanding their relationship is key to determining their domains. In this comprehensive guide, we'll explore what logarithmic functions are, how to find their domains, and work through various examples to solidify your understanding.

What is a Logarithmic Function?

A logarithmic function is the inverse of an exponential function. In simpler terms, if an exponential function tells you what you get when you raise a base to a certain power, a logarithmic function tells you what power you need to raise the base to in order to get a certain number.

Mathematically, if we have an exponential equation like:

b^y = x

where b is the base, y is the exponent, and x is the result, we can rewrite this in logarithmic form as:

log_b(x) = y

This reads as "the logarithm of x to the base b is y." The base b must be a positive real number not equal to 1, and x must be a positive real number. This constraint on x is crucial when determining the domain of a logarithmic function.

Key Components:

• Base (b): The base of the logarithm. It must be a positive real number and not equal to 1.
• Argument (x): The value inside the logarithm, also known as the logarithmand. This is the value for which you're finding the logarithm.
• Logarithm (y): The result of the logarithmic function, which represents the exponent to which the base must be raised to obtain the argument.

Understanding Domain Restrictions

The domain of a function is the set of all possible input values (x-values) for which the function is defined and produces a real number output. For logarithmic functions, there's a critical restriction on the domain:

The argument of a logarithm must be strictly greater than zero.

In other words, we can only take the logarithm of positive numbers. We cannot take the logarithm of zero or a negative number. This is because there's no power to which you can raise a positive base to get zero or a negative number.

Therefore, to find the domain of a logarithmic function, we need to ensure that the expression inside the logarithm is always positive. This often involves setting up an inequality and solving for x.

Steps to Find the Domain of a Logarithmic Function

Here's a step-by-step guide to finding the domain of a logarithmic function:

1. Identify the Argument: Locate the expression inside the logarithm. This is the part that follows the "log" and the base (if specified). For example, in log(x+2), the argument is x+2.

2. Set Up the Inequality: Set the argument greater than zero. This reflects the restriction that the argument of a logarithm must be positive. So, in the example above, we would write x+2 > 0.

3. Solve the Inequality: Solve the inequality for x. This will give you the range of values for x that satisfy the condition that the argument is positive. In our example, solving x+2 > 0 gives us x > -2.

4. Express the Domain: Write the domain in interval notation. Interval notation uses parentheses and brackets to indicate the range of values included in the domain.

   • Parentheses () indicate that the endpoint is not included in the domain.
   • Brackets [] indicate that the endpoint is included in the domain.
   • Infinity ∞ and negative infinity -∞ are always enclosed in parentheses because they are not actual numbers.

   In our example, since x > -2, the domain is (-2, ∞). This means that any value of x greater than -2 is in the domain of the function.

Examples of Finding the Domain of Logarithmic Functions

Let's work through several examples to illustrate the process of finding the domain of logarithmic functions.

Example 1: f(x) = log(x - 3)

1. Identify the Argument: The argument is x - 3.

2. Set Up the Inequality: x - 3 > 0

3. Solve the Inequality: x > 3

4. Express the Domain: (3, ∞)

Example 2: g(x) = log_2(5 - x)

1. Identify the Argument: The argument is 5 - x.

2. Set Up the Inequality: 5 - x > 0

3. Solve the Inequality:

   • -x > -5
   • x < 5 (Remember to flip the inequality sign when multiplying or dividing by a negative number.)

4. Express the Domain: (-∞, 5)

Example 3: h(x) = ln(2x + 1)

Note: ln(x) represents the natural logarithm, which is a logarithm with base e (Euler's number, approximately 2.71828).

1. Identify the Argument: The argument is 2x + 1.

2. Set Up the Inequality: 2x + 1 > 0

3. Solve the Inequality:

   • 2x > -1
   • x > -1/2

4. Express the Domain: (-1/2, ∞)

Example 4: k(x) = log((x + 4)(x - 2))

1. Identify the Argument: The argument is (x + 4)(x - 2).

2. Set Up the Inequality: (x + 4)(x - 2) > 0

3. Solve the Inequality: This involves solving a quadratic inequality. We need to find the intervals where the product (x + 4)(x - 2) is positive.

   • Find the critical points: Set each factor equal to zero and solve for x:
     • x + 4 = 0 => x = -4
     • x - 2 = 0 => x = 2
   • Create a sign chart: Divide the number line into intervals using the critical points: (-∞, -4), (-4, 2), and (2, ∞).
   • Test each interval: Choose a test value within each interval and plug it into the inequality (x + 4)(x - 2) > 0 to see if it's true or false.
     • Interval (-∞, -4): Let x = -5. (-5 + 4)(-5 - 2) = (-1)(-7) = 7 > 0. The inequality is true.
     • Interval (-4, 2): Let x = 0. (0 + 4)(0 - 2) = (4)(-2) = -8 > 0. The inequality is false.
     • Interval (2, ∞): Let x = 3. (3 + 4)(3 - 2) = (7)(1) = 7 > 0. The inequality is true.
   • Identify the intervals where the inequality is true: The inequality is true for (-∞, -4) and (2, ∞).

4. Express the Domain: (-∞, -4) ∪ (2, ∞) (The symbol ∪ represents the union of two intervals.)

Example 5: m(x) = log(-x^2 + 5x - 4)

1. Identify the Argument: The argument is -x^2 + 5x - 4.

2. Set Up the Inequality: -x^2 + 5x - 4 > 0

3. Solve the Inequality: It's often easier to work with a positive leading coefficient, so multiply both sides by -1 and flip the inequality sign:

   • x^2 - 5x + 4 < 0
   • Factor the quadratic: (x - 1)(x - 4) < 0
   • Find the critical points:
     • x - 1 = 0 => x = 1
     • x - 4 = 0 => x = 4
   • Create a sign chart: Divide the number line into intervals: (-∞, 1), (1, 4), and (4, ∞).
   • Test each interval:
     • Interval (-∞, 1): Let x = 0. (0 - 1)(0 - 4) = (-1)(-4) = 4 < 0. The inequality is false.
     • Interval (1, 4): Let x = 2. (2 - 1)(2 - 4) = (1)(-2) = -2 < 0. The inequality is true.
     • Interval (4, ∞): Let x = 5. (5 - 1)(5 - 4) = (4)(1) = 4 < 0. The inequality is false.

4. Express the Domain: (1, 4)

Example 6: n(x) = log(x^2 + 1)

1. Identify the Argument: The argument is x^2 + 1.

2. Set Up the Inequality: x^2 + 1 > 0

3. Solve the Inequality: Notice that x^2 is always non-negative (zero or positive) for any real number x. Therefore, x^2 + 1 is always greater than or equal to 1, which is always greater than 0. So, the inequality x^2 + 1 > 0 is true for all real numbers.

4. Express the Domain: (-∞, ∞)

Common Mistakes to Avoid

• Forgetting the Argument Restriction: The most common mistake is forgetting that the argument of a logarithm must be greater than zero. Always remember to set up the inequality and solve for x.

• Incorrectly Solving Inequalities: Pay close attention to the rules for solving inequalities, especially when multiplying or dividing by a negative number (remember to flip the inequality sign).

• Confusing Interval Notation: Make sure you understand the difference between parentheses and brackets in interval notation. Parentheses indicate that the endpoint is not included, while brackets indicate that it is included.

• Not Considering All Critical Points: When dealing with quadratic or other polynomial inequalities, make sure you find all the critical points (where the expression equals zero) and use them to divide the number line into intervals.

• Assuming All Logarithmic Functions Have Restricted Domains: While most logarithmic functions have restricted domains, some, like log(x^2 + 1) in the example above, are defined for all real numbers. Always go through the steps to determine the domain.

Domain of Log Function: FAQ

Q: What is the domain of the natural logarithm, ln(x)?

A: The natural logarithm, ln(x), has a base of e (Euler's number). Its domain is (0, ∞), meaning it's defined for all positive real numbers.

Q: Why can't the argument of a logarithm be zero?

A: If the argument of a logarithm were zero, it would imply that b^y = 0 for some base b. However, no matter what power you raise a positive base to, the result will never be zero.

Q: Why can't the argument of a logarithm be negative?

A: If the argument of a logarithm were negative, it would imply that b^y = negative number for some base b. If b is a positive real number, then no real number y will satisfy the expression.

Q: What if I have a logarithmic function inside another function?

A: In such cases, you need to consider the domain restrictions of both the logarithmic function and the outer function. For example, if you have sqrt(log(x)), you need to ensure that log(x) is non-negative (greater than or equal to zero) because it's inside a square root, and that x is positive because it's the argument of a logarithm.

Q: Can the base of a logarithm be negative?

A: No, the base of a logarithm must be a positive real number not equal to 1. This is a fundamental requirement for logarithms to be well-defined.

Conclusion

Finding the domain of a logarithmic function is a crucial skill in mathematics. By understanding the fundamental restriction that the argument of a logarithm must be positive and following the steps outlined in this guide, you can confidently determine the domain of various logarithmic functions, including those with more complex arguments. Remember to practice with different examples to solidify your understanding and avoid common mistakes. Mastering this concept will significantly enhance your ability to work with logarithmic functions in various mathematical contexts.